As usual, I'll post a traditionalists topic on test day. Some readers might be tired of all those music posts I've written lately, while others are grateful that with so many music posts, there's less time for endless traditionalists posts.
But actually, the reason I haven't had as many traditionalists posts lately isn't music. It's because our primary traditionalist, Barry Garelick, hasn't posted in four weeks -- and that means that his main commenter, SteveH, has been silent as well. Today's post, then, will be a hodgepodge of what other traditionalists besides Garelick and SteveH are currently discussing.
Let's start with the Joanne Jacobs website, where the traditionalist Bill often comments. Here is the most recent Jacobs post with a Bill comment:
http://www.joannejacobs.com/2018/06/22-of-pomona-students-are-disabled/
“As many as one in four students at some elite U.S. colleges are now classified as disabled, largely because of mental-health issues such as depression or anxiety,” reports Douglas Belkin in the Wall Street Journal. Under federal law, that entitles them accommodations such as more time to take exams or a private, distraction-free testing room.
At Pomona, 22% of students were considered disabled this year, up from 5% in 2014. . . . At Hampshire, Amherst and Smith colleges in Massachusetts and Yeshiva University in New York, one in five students are classified as disabled. At Oberlin College in Ohio, it is one in four. At Marlboro College in Vermont, it is one in three.
Anyway, here's what Bill wrote:
Bill:
It is amazing the actual lack of knowledge that high school graduates have anymore. Due to self-esteem mantra and ‘participation trophies’, they get praised for ‘trying’, but not actually succeeding.
As a result, these students usually fail in college, and struggle with employment as they all want to start at the top in jobs, instead of working their way up the ladder.
Employers are having a difficult time finding qualified workers, mostly due to a lack of skills, most of which should have been learned in public school.
Be prepared for more issues as students are handed diplomas they didn’t actually earn, and when they fail in college (and on the job), perhaps they’ll understand the importance of getting an actual education.
In other words, Bill implies that the real "disabilities" are "lack of knowledge" and "lack of skills." I point out that nowhere in this comment does Bill say that it's a lack of math skills. But another commenter in the thread, "lee," does specifically mention math:
lee:
Re: lack of life skills amongst students –
Once I tutored a community college student who was having difficulty working with negative numbers, I tried framing the situation using a money context. She still was thoroughly confused, and I encouraged her to think about negative numbers as money owed/money going out, and positive numbers as getting paid/money coming in. Surely she would have enough familiarity with basic personal finance to grasp the parallel. Nope, no dice – she freely admitted that her parents handled all that for her, so she didn’t have to worry about budgeting and so on.
Sheesh. If she hasn’t learned anything by now, she’s due for a world of hurt.
As it turns out, Joanne Jacobs just spent a week and a half in Europe. In her place, she had Darren Miller of "Right on the Left Coast" post a few articles. Once again, "left coast" means California, and "right" means right-wing or conservative. I often forget about Miller's own blog, which I consider to be more political than traditionalist. But seeing his posts on the Jacobs website reminded me to check out his own blog -- where I found a recent Common Core post:
http://rightontheleftcoast.blogspot.com/2018/05/common-core-disaster.html
I'm not qualified to comment on the Common Core English standards, but I'm qualified enough to comment on the CC math standards. I stand by what I've said since those math standards were first published--they're not as rigorous as California's 1997 standards. Nor are they as clear.
Essentially, they're crap compared to what we had.
But if students are learning more, they're not crap, right? The proof is in the pudding, right? Why yes, yes it is, and California's kids aren't doing better in math:
At this point Miller links to a Hoover article -- which focuses on the fact that the old California standards encouraged eighth grade Algebra I, while the Common Core discourages it. Then he reposts his first comment from the Hoover article:
8th grade Algebra 1 is one valuable marker, and you rightly point out that California's students have slipped on that metric since California adopted Common Core. But wouldn't a better standard be "how well are students learning?" How were California students performing on benchmark tests before CCSS, and how does their performance now compare?
OK, I'll buy that. Traditionalists are very obsessed with eighth grade Algebra I and senior-year AP Calculus, and so maybe it is better just to look at overall math performance, not two specific courses at two specific grade levels.
One of Miller's own commenters calls herself "Auntie Ann":
Auntie Ann:
I wrote this almost exactly 4 years ago. It was part of a longer post on CC and the problems around it: Common Core:
However, as I said above, the Common Core is not a selective-college or STEM-ready standard. Poor and disadvantaged students, or students with poorly educated parents, will be completely reliant on the schools to give them their math curriculum. On the other hand, students in the middle-class or better, and students with well-educated parents will get more.
The family in Santa Monica that I mentioned earlier will not simply have their daughter twiddle her thumbs for the next two years until her peers catch up with her and she can finally be allowed to take algebra. The family will hire the tutors necessary to make sure that their child continues to stay on the track she is on now.
There are six Kumon centers within five miles of where I’m sitting.
OK, I don't need to cut and paste the rest of this post, since we all know where it's going. Auntie Ann sounds almost exactly like SteveH here -- both emphasize how Common Core isn't pro-STEM since it discourages middle school Algebra I. And both commenters point to Kumon tutoring as a way to circumvent the Core.
Notice that Auntie Ann mentions a girl in Santa Monica (which is also right here in Southern California) who's forced by her district to "twiddle her thumbs for the next two years." This implies that according to Auntie Ann, the girl ought to be enrolled in seventh grade Algebra I, eighth grade Geometry, and freshman Algebra II -- and that she would have been had it not been for the Core.
While the Common Core can rightly be blamed for one of those years, the standards aren't the reason why she can't take seventh grade Algebra I. Without the Core, she could indeed take Algebra I as the only seventh grader in a class of eighth graders -- but then what would she take in eighth grade? It's always problematic to accelerate a student who's in the highest grade level on that campus. She can't take Geometry in eighth grade because she'd be in a class of only one student.
By the way, here's a disclaimer -- I myself took Geometry as a young eighth grader. But I had the advantage of attending a school that spanned Grades 7-12. Thus it was easy for me to be the only eighth grader in a class of high school students. (The specific district mentioned by Auntie Ann actually does have a 7-12 school, but it's in another city. Most likely, the girl's house is closer to the Kumon center than to the 7-12 school.)
By the way, here's one more related article posted at Darren Miller's blog:
It must be that time of year again, the time when we start seeing a new rash of articles on why high school students don't need calculus. Sigh.
Thousands of American high-school students on Tuesday will take the Advanced Placement calculus exam. Many are probably dreading it, perhaps seeing the test as an attempt to show off skills they will never use. What if they’re right?
Miller continues:
I love statistics. I teach statistics. I'd love to have even more students learning statistics. But I'm not going to denigrate the study of calculus just to inflate statistics enrollment. They're both important courses, and they teach different skills and different ways of thinking. One isn't "better" than the other. One is more practical, at least for the "common person", but that isn't enough of a reason to diminish the study of calculus.
Once again, I don't care whether some WSJ journalist thinks Calculus is necessary or not. I care about whether the students actually sitting in the Calculus classes taking the AP test think Calculus is necessary or not? Again, I ask, in which class are the questions "Why do we have to learn this?" or "When will we use this in real life?" more likely to be asked (again, by students, not WSJ journalists), Calculus or Statistics? And what can we, as teachers, do to make the question "When will we use this in real life?" disappear from that class?
And again, Auntie Ann is the first commenter. (Is Ann the "SteveH" of Miller's blog?)
Auntie Ann:
Apparently, California has decided that no one needs calc. From 59% taking Algebra in 8th grade--with the opportunity of high school calc--to 19% in just 3 years. From blacks accelerating gains rapidly, to a complete crash and burn in 3 years
California’s Common Core Mistake by Williamson M. Evers, Ze’ev Wurman
>> But with the Common Core standards, this progress began to stall. Common Core expects Algebra I in the ninth grade. That threw a monkey wrench in California’s longstanding effort introducing the math class to students earlier. As seen in the chart below, in the four years under Common Core, the number of eighth graders taking Algebra I in California dropped precipitously to 19 percent in 2017, taking California back to where it was around 1999, when early algebra taking was the privilege of the elite. And while all demographic groups lost ground, the loss for Latino and African American students was much deeper than for white and Asian Americans. <<
Here Auntie Ann quotes Ze'ev Wurman, another major traditionalist. And oops, race is mentioned in this post -- it's inevitable that traditionalists lead to tracking, and tracking leads to race.
Meanwhile, I've been thinking more about the Path Plan that I've mentioned in previous posts. This is based on a plan that my old elementary school tried to implement when I was a young student.
Let's review what exactly the Path Plan is. Students are divided into classes based not on grade level, but a new type of level called a "path." The paths do correspond roughly to grades, as follows:
Early Learning Path: Grade K
Primary Path: Grades 1-2
Transition Path: Grades 3-4
Preparatory Path: Grades 5-6
Students are placed into paths based on their reading level. Thus an advanced second grader might be on the Transition Path and an advanced fourth grader might be on the Preparatory Path. Meanwhile, a below basic third grader might be on the Primary Path and a below basic fifth grader might be on the Transition Path. Thus one purpose of the Path Plan is that is serves as a mild form of tracking.
The other purpose of the Path Plan is that it allows teachers to specialize. In the version of the Path Plan that I've posted, all paths learn ELA in homeroom until everyone has had recess. Then the kids on each path attend classes with different teachers:
Primary Path: math only
Transition Path: math, elective
Preparatory Path: math, elective, science
The idea is that on the higher paths, only some of the teachers must teach math. This allows the strongest math teachers to cover the subject, while weaker math teachers can teach another subject.
But there's a problem with this sort of specialization, according to the following Chalkbeat article:
https://www.chalkbeat.org/posts/ny/2018/05/17/as-nyc-encourages-more-elementary-teachers-to-specialize-in-math-new-research-shows-the-strategy-could-hurt-student-learning/
But two new pieces of research suggest that the city’s approach — a crucial piece of its plan to get all students ready for algebra in middle school — might be misguided. The studies, which examine “departmentalization” efforts in other places, raise questions about whether the rapidly expanding city program that encourages schools to departmentalize fifth-grade math could be doing more harm than good.
(Yes, it's ironic that this is all about preparing students for Algebra I in middle school -- the opposite of what we're seeing in California.)
Anyway, the link above discusses some of the problems with elementary education. We know that elementary teachers are expected to teach multiple subjects. And more importantly, prospective teachers often choose the field because they like little kids -- not because they like math.
Indeed, the article is all about specialization in fifth grade math. This is the year that students are expected to learn about, among other things, fractions. But many fifth graders tell themselves, "I hate fractions" -- then these same kids grow up to become fifth grade teachers who must cover fractions!
So the idea seems sound that it's better for fifth graders to learn fractions from a specialist who is comfortable with fractions. Yet that's not what the study in the article reveals. Instead, it reveals that it's more important for teachers to know and have a deep understanding of the students -- which is easier if the students have a single teacher the entire day -- than it it for the teachers to know and have a deep understanding of fractions. As the cliche goes, "teachers teach students, not subjects" -- and this is especially true in elementary school.
If this study is accurate, then this destroys the Path Plan. The study implies that it's better even on the Preparatory Path, the path that contains fifth grade (much more the lower paths) for students to learn math from their homeroom teacher rather than a math specialist.
Why did I post the Path Plan, anyway? It all goes back to another Joanne Jacobs poster, momof4. She often writes about homeschooling, and that many parents choose to homeschool because they disagree with how public schools are organized.
Earlier today, momof4 participated in a thread that suggested that one reason to homeschool is the recent school shootings:
http://www.joannejacobs.com/2018/06/safety-fears-encourage-homeschooling/
But momof4 provides other reasons to homeschool. She begins by telling a story about a student who is chronologically of high school age, yet isn't even toilet trained.
momof4:
I would have some shreds of respect for teachers’ unions if they would use their undoubted political power to demand a safe and orderly school environment and to demand appropriate education for the kids willing to be taught – separately, according to academic need.
And in other threads, she writes "academic need by subject." She implies that if public schools (and their unions, apparently) would push for this, then there would be fewer homeschoolers.
The problem, of course, occurs when a student is above average in one subject (say math), while below average in another subject (say reading). Even under a simple tracking system, the teacher with the below average kids teaches basic material in both reading and math. A student who is below in reading but above in math would have to go to another teacher during math time -- that is, they'd have more than one teacher during the day. Yet according the Chalkbeat study, students with more than one teacher during the day don't fare as well as those with only a single teacher.
In other words, homeschooling has an advantage that doesn't scale up. If a mother has a child who is above grade level in one subject but below in another, it's easy for her to give the child appropriate material in both subjects, even while the child has only one teacher (the mother). In fact, this is possible even if she has more than one child -- say four (as the username "momof4" implies). It's easy for her to provide for all the subject/grade level combinations, since there are only four of them.
And another poster in the thread, Homeschooling Granny, tells us that even specialization is possible for homeschooling:
Homeschooling Granny:
The adult in the home cares more about the kid than any teacher – and that is NOT to suggest that teachers don’t care, just that the parent has more at stake. Given their higher level of involvement with their kids, I wouldn’t assume that parents are less effective than schools. For example, in my experience, parents are quick to switch curriculums if their kids struggle with one, while a teacher is likely to be locked into the chosen curriculum. And parents are quick to form alliances to take advantage of each others’ specialties, creating cooperatives.
For example, if momof4 is stronger at reading and Homeschooling Granny is stronger at math, then HSG can take mo4's children while mo4 can take HSG's grandchildren. Since the total number of children is still small, the negative effect of specialization (according to Chalkbeat) doesn't occur -- but once again, this can't scale up to a full-sized public classroom.
Notice that HSG mentions something else that doesn't scale up -- "parents are quick to switch curricula if their kids struggle with one." Even if we had a system where different teachers at the same school are teaching different curricula, suppose it's realized on Day 128 that one curriculum isn't working for a student -- chances are that switching on Day 128 isn't convenient for anyone involved, teacher or student, unless there are no or very few other students (as in homeschool). And the perfect school that fits your son's needs might be miles away from the perfect school that fits your daughter's needs. In short, homeschoolers say that they'd return to the public schools if there could
Returning to Chalkbeat, there's something that might help solve this problem. I noticed that the first Chalkbeat article links to a second article:
https://chalkbeat.org/posts/us/2018/03/21/elementary-school-teachers-sometimes-follow-a-class-of-students-from-year-to-year-new-research-suggests-thats-a-good-idea/
Now, a new study seems to confirm Van Duzer’s experience. Students improve more on tests in their second year with the same teacher, it finds, and the benefits are largest for students of color.
I've mentioned "looping" before in connection with Waldorf schools, where students have a single teacher from first all the way to eighth grade. I'm not promoting such an extreme looping plan, but again, the idea is that students learn more when they have a deeper relationship with the teacher.
And so we can ameliorate the disadvantages of specialization by adding looping to the Path Plan. (I point out that looping wasn't part of the Path Plan as my old elementary school followed it.)
Basically, the idea is that students can have a single homeroom teacher the entire time they are on a given path. Again, the normative student spends two years in each path (Grades 1-2 on Primary, Grades 3-4 on Transition, Grades 5-6 on Preparatory), so that means that a student has the same homeroom teacher for two years.
And notice that a Primary Path student has only one class outside of homeroom -- math. The idea is that the Primary Path teachers can be paired up -- one teaches first grade math while the other teaches second grade math. After recess, the first grade math teacher takes all the first graders in both classes and the second grade teacher takes all of the second graders in both classes.
For Transition (or Preparatory) Path, we can pair up the teachers again. This time, one teacher in each pair teaches both third and fourth (or both fifth and sixth) grade math while the other teacher is doing art or some other elective. At my old elementary school, there was an exploratory wheel where students rotate and learn a different subject each trimester. This increases the number of teachers a student has -- but it keeps the number of teachers in tested subjects (ELA and math) to a minimum, so this shouldn't have a negative impact on test scores.
I actually found the Chalkbeat links on the CCSSIMath Twitter page -- this is the one where the pinned tweet shows students learning the circle area formula in fifth grade in Japan, as opposed to high school Geometry. I pointed out that in the U.S., we want to delay as much math as possible at least to the years where specialists teach math. We wouldn't want to force fifth grade teachers who aren't comfortable with fractions to teach pi r^2.
So if we really want our fifth and sixth graders to have a more rigorous curriculum, one in which they can solve the Japan problem and others posted on the CCSSIMath Twitter page, then we really need some sort of specialization. The Path Plan in this post explains how we can have specialization yet allow the relationship between teacher and student to develop.
By the way, today CCSSIMath tweeted another "sixth grade problem":
https://twitter.com/CCSSIMath/status/1001577636542472193
As usual, I don't see anyway to solve this other than coordinates. We could let the original square be a unit square with vertices at (0, 0), (1, 0), (1, 1), and (0, 1). But this will introduce many fractions, and we can avoid these by letting the vertices be (0, 0), (8, 0), (8, 8), and (0, 8).
This will allow us to find the vertices of the triangle whose area we are to find. One vertex is clearly at (0, 8). The second is the midpoint of
So we need the area of a triangle with vertices (0, 8), (2, 4), and (5, 6).
To do so, we need its base and height. Let (0, 8) to (2, 4) be the base -- since (0, 8) contains a 0, it's easier to find this distance as sqrt(20) or 2sqrt(5). The slope of the line joining these two points is -2, and so the height is length of the line of slope 1/2 passing through (5, 6).
y - 6 = 1/2(x - 5)
y - 6 = (1/2)x - 5/2
y = (1/2)x + 7/2
The base, with slope -2 and y-intercept 8, must have equation y = -2x + 8
y = (1/2)x + 7/2
y = -2x + 8
(1/2)x + 7/2 = -2x + 8
(5/2)x = 9/2
x = 1.8
y = -2(1.8) + 8
y = 4.4
So the height is the distance from (1.8, 4.4) to (5, 6), which is sqrt(3.2^2 + 1.6^2) = sqrt(5 * 1.6^2), which can be written as 1.6sqrt(5).
Finally, the area of the triangle is (1/2)(2sqrt(5))(1.6sqrt(5)) = 1.6(5) = 8. The area of the square is 64, and so the ratio of the triangle's area to the square's is 8/64 = 1/8.
Now that I see that the answer is 1/8, I wonder whether it's possible for a sixth-grader to obtain this answer without using algebra at all. For example, the line
But then CCSSIMath adds another step to this problem -- a blue line from A to (8, 8) is drawn in. It divides the goal triangle into two regions, 1 and 2 -- and we're asked to find the ratio of the area of Region 1 to that of Region 2.
I notice that this newly drawn line is perpendicular to the segment from (0, 8) to (2, 5), and thus Region 1 is a right triangle. I hope this will help us -- let's just find the area of Region 1.
The blue line has slope 1/2 and y-intercept 4, hence equation y = (1/2)x + 4. Let's find out where it intercepts the two sides of the triangle:
y = (1/2)x + 4
y = -2x + 8
(1/2)x + 4 = -2x + 8
(5/2)x = 4
x = 1.6
y = -2(1.6) + 8
y = 4.8
y = (1/2)x + 4
y = (-2/5)x + 8
(1/2)x + 4 = (-2/5)x + 8
0.9x = 4
x = 40/9
y = (-2/5)(40/9) + 8
y = 56/9
So the points of intersection are (1.6, 4.8) and (40/9, 56/9). Since this is a right triangle, we only need to find the length of its legs and multiply:
Distance from (0, 8) to (1.6, 4.8) = sqrt(1.6^2 + 3.2^2) = 1.6sqrt(5)
Distance from (1.6, 4.8) to (40/9, 56/9) = sqrt((128/45)^2 + (64/45)^2) = (64/45)sqrt(5)
Area of Region 1 = (1/2)(1.6sqrt(5))((64/45)(sqrt(5)) = 256/45
Area of Region 2 = 8 - 256/45 = 104/45
So the desired ratio is 256/104 = 32/13. I don't see any simpler way to obtain this ratio. It took me about an hour to solve this so-called "sixth grade problem."
Oh, and since I've already mentioned race in this post, what about politics? Tomorrow is the California gubernatorial primary. I've already chosen which candidate I'll support, but I choose not to reveal it on the blog.
I will say that the three main Democratic candidates are Gavin Newsom, Antonio Villaraigosa, and John Chiang. The California "jungle primary" makes it possible that two Democrats will advance to November rather than one from each major party. According to most sources, Villaraigosa in more pro-charter, while Newsom is more pro-union. Let's try to keep this post politically neutral -- I already wrote about "Right on the Left Coast" and Darren Miller. Since he's conservative, I expect him to support one of the Republican candidates. Of the GOP candidates, the one most likely to advance to November is John Cox. None of the Republicans has said much about education in the days leading up to the primary.
I never posted the second semester final last year to the Great Post Purge of 2017. Therefore, this is what I wrote two years ago about today's test:
This is finals week at the school district whose schedule I'm following on the blog. And so today I am posting my version of the second semester final exam.
As usual, let me give my rationale for choosing these particular questions. When I wrote this final, I wanted it to serve not only as an in-classroom final, but what my vision of an ideal Common Core test, like PARCC or SBAC, should look like.
I've talked several times about the traditionalists who prefer that test questions focus more on content and less on labels. The questions at the end of each chapter of the U of Chicago are divided into four sections, Skills, Properties, Uses, and Representations (SPUR). So we conclude that the traditionalists prefer tests that are heavy on Skills (where most of the content is), and light on Properties (where most of the labels are).
I don't agree completely with the traditionalists here -- especially not in Geometry class. Geometry, after all, is all about proofs, and the reasons that appear in proofs are labels and properties. So if one isn't learning about labels and properties, then one isn't really doing Geometry.
[2018 Update: By the way, here is what traditionalist Auntie Ann wrote about Geometry in her old link from four years ago:
5. In the only direct complaint against the actual standards I have seen, supposedly, the high school geometry standards came out of nowhere and embrace an odd view of geometry. I don't know if this is true or not, but I do know that the classic geometry of proofs and theorems and corollaries has been dying for a long time. Geometry was my favorite math class, and I loved the proofs-based course. If it has been dying for a while, it is hard to attribute that to CC.
7. This includes the word-heavy explanations required in K-12 math today, and the belief that if you can’t explain something in words, then you don’t really understand it. Showing your work used to be enough to show understanding: if you could show the steps you took, you already showed your understanding. Now, even simple tasks have to be explained in complete sentences.
Notice that Auntie Ann's #7 seems to contradict her #5 -- the proofs of Geometry really are explanations in words of why a theorem is true. Then again, she proceeds to give arithmetic as an example of what she means by #7. Thus apparently, Ann has no problem with word-heavy explanations (proofs) in Geometry -- only in arithmetic (probably up to Algebra I).]
A test that selects from the questions in the U of Chicago text would naturally have mostly Properties and Representation questions, and this is what I started to write. But one traditionalist argument for having more Skills than Properties questions is that with a Skills-based test, students who have the necessary Skills can take the test cold, without having to study for a long time, and still get an excellent grade. But a test that contains many labels and properties would require even the smartest students to spend time learning the particular names of the labels and properties. This is significant considering that one major argument against standardized tests like the Common Core tests is that they require so much time for test prep.
I spent lots of time on this blog preparing for the PARCC test -- not my final exam. Yet I didn't want my test to be just PARCC problems. And so I took questions from the U of Chicago text -- and since I didn't post test review for these question on the blog, they ended up being Skills questions, just as the traditionalists desire.
So here's how I wrote the test. This is a cumulative exam covering the whole text. But it was hard for me to find some good problems for Chapters 1 and 2, and I did just post a review sheet last month for some of the angle theorems from Chapter 3, so I began with Chapter 3. I decided to include numbered questions from the text that were multiples of five, starting with Question 5 and stopping at the end of the Skills section. For Chapter 3, there are six questions that would be included, Questions 5, 10, 15, 20, 25, and 30. But I had to drop Questions 10 and 30 because the particular skill for those questions involve drawing, which isn't easy to do on either a multiple-choice final or a computerized Common Core test.
Here is the chapter breakdown: for Chapter 3, I included four questions, but for Chapter 4, I included just one question. For Chapter 5, I included four questions, but for Chapter 6, I included just one question again. We notice that Chapters 4 and 6, where the transformations are taught, have very few Skills questions, since the main skill in both chapters is drawing the images, and I've already decided to drop all drawing questions. This is in accord with the traditionalist distaste for the Common Core transformations like reflections and translations. But Chapter 7 has just two included questions -- for questions on SSS, SAS, and ASA are also just Property questions.
Chapter 8 has the most included questions, with a whopping ten of them. Seven of these questions are from the Skills section. But after I wrote this test, I've having second thoughts about these. Many of these questions are not straightforward. For example, students are asked to find the perimeter of a square given its area or vice versa, as opposed to finding either of them given the side length. Now traditionalists like these types of problems because they require students to think deeply about the problem -- and I agree, but only up to a point. I have no problem with some of the questions requiring students to think outside the box, but when every question is this difficult, students will eventually become frustrated. But unfortunately, I ended up choosing the multiples of five, and these just happen to be the more difficult problems.
No matter what anyone else says, I want to include some problems from the Uses section, since I still want to demonstrate how math can be applied to the real world. So this means that I include questions 50, 55 and 60 from the Uses section of Chapter 8.
Chapter 9 is a tough chapter, since we covered Chapter 9 only briefly so we could get to Chapter 10. I included three questions (one from Uses) for Chapter 9. Chapter 10 is, of course, a big chapter, and so I included six questions (two from Uses) for this chapter.
Chapter 11, on coordinate geometry contains no Skills problems at all. Dr. David Joyce criticizes coordinate geometry, and so I include only two Uses questions from this chapter. From Chapter 12 I included five questions, with two of them from the Uses section.
Chapter 13 contains very few Skills or Uses questions -- it's a chapter focusing mainly on Properties, just like Chapter 2. So I included no questions from this chapter -- and recall that Chapter 13 will be broken up for my curriculum next year. From Chapter 14 I included six questions, with one of them from the Uses section. Since I covered Chapter 15 only briefly, I was only able to include one question from this chapter -- otherwise Chapter 15 would be a great Skills-based chapter.
This leaves five questions from the PARCC Practice test. I decided to continue the pattern and stick to multiples of five, so I included questions 10, 15, 20, 25, and 30. As we expect for PARCC questions, of course these are mostly Properties questions. These are already set up to be multiple choice -- still I had to set up the questions from the U of Chicago text so that they could be multiple choice as well.
Of course, I set up the questions to be multiple-choice for the purposes of the final. If this really were a computer-based exam, then I would have more free-response questions -- especially those requiring students to enter only a numerical answer.
Notice that Representations has been completely shut out of this test -- and Representations includes graphs and coordinate geometry. One problem with graphing questions on the computer is that they often require students to drag the graph to the correct location -- and this confuses them. I believe that there should be more graphing questions, but it's not clear to me how to make them so that more students can draw them on the computer easily. The only question that involves a graph is officially a Skills-based question from Chapter 8 -- students are to estimate the area of an irregular region.
I still like the idea of a computer-based test, though. Many people say that they oppose Common Core because it's "one size fits all." But the whole point of a computer-adaptive test like the SBAC is to avoid being "one size fits all" -- the same test for every student. I can easily imagine a computerized test asking a question such as to find the area of a square given its perimeter. A student who answers this incorrectly (say, by simply squaring the perimeter). can get an easier question such as to find the area of a square given its side length. Those who answer correctly, on the other hand, can get more difficult questions such as to find the area of a circle given its diameter or circumference, then move on to difficult volume questions, and so on.
Indeed, students who get many questions right could move on to some above-grade-level questions, if time allows. Unfortunately, I doubt that the actual SBAC does this. So SBAC fails to use the full power of having a computer-adaptive test. I wonder whether more traditionalists would be in favor of a computer-adaptive test like the SBAC if students could jump to above-grade-level questions.
Here are the answers to my posted final exam:
CAADA ACDDD ABADB ADCAC ABCBC BAADC ACACB ADBDA ADBCD ACCDC
Once again, I don't post a Form B for this exam.
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